A ball is kicked with an initial height of 0.75 meters and initial upward velocity of 22 meters/second. This inequality represents the time, [tex]$t$[/tex], in seconds, when the ball's height is greater than 10 meters.
[tex]$-4.9 t^2+22 t+0.75>10$[/tex]
The ball's height is greater than 10 meters when [tex]$t$[/tex] is approximately between seconds.
Answer (2)
The ball's height is greater than 10 meters when time t is approximately between 0.47 seconds and 4.02 seconds. We found this by solving the inequality related to the ball's height using the quadratic formula. The solution indicates that during this interval, the height of the ball exceeds 10 meters.
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Rewrite the inequality 10"> − 4.9 t 2 + 22 t + 0.75 > 10 as 0"> − 4.9 t 2 + 22 t − 9.25 > 0 .
Use the quadratic formula to find the roots of the equation − 4.9 t 2 + 22 t − 9.25 = 0 .
Calculate the discriminant: D = 2 2 2 − 4 ( − 4.9 ) ( − 9.25 ) = 302.7 .
Approximate the roots: t 1 ≈ 0.47 and t 2 ≈ 4.02 . The solution is the interval ( 0.47 , 4.02 ) , so the final answer is 0.47 , 4.02 .
Explanation
Understanding the Problem We are given the inequality 10"> − 4.9 t 2 + 22 t + 0.75 > 10 , which represents the time t in seconds when the ball's height is greater than 10 meters. Our goal is to find the interval of time t for which this inequality holds true.
Rewriting the Inequality First, let's rewrite the inequality by subtracting 10 from both sides: 0"> − 4.9 t 2 + 22 t + 0.75 − 10 > 0 0"> − 4.9 t 2 + 22 t − 9.25 > 0
Using the Quadratic Formula Now, we need to find the roots of the quadratic equation − 4.9 t 2 + 22 t − 9.25 = 0 . We can use the quadratic formula: t = 2 a − b ± b 2 − 4 a c
where a = − 4.9 , b = 22 , and c = − 9.25 .
Calculating the Discriminant Let's calculate the discriminant, D = b 2 − 4 a c : D = 2 2 2 − 4 ( − 4.9 ) ( − 9.25 ) = 484 − 181.3 = 302.7
Finding the Roots Now, we can find the roots t 1 and t 2 : t 1 = 2 ( − 4.9 ) − 22 − 302.7 = − 9.8 − 22 − 302.7
t 2 = 2 ( − 4.9 ) − 22 + 302.7 = − 9.8 − 22 + 302.7
Approximating the Roots Using a calculator, we find the approximate values of the roots: t 1 ≈ − 9.8 − 22 − 17.4 ≈ − 9.8 − 39.4 ≈ 4.02 t 2 ≈ − 9.8 − 22 + 17.4 ≈ − 9.8 − 4.6 ≈ 0.47
Determining the Time Interval Since the coefficient of the t 2 term is negative ( − 4.9 ), the parabola opens downward. Therefore, the inequality 0"> − 4.9 t 2 + 22 t − 9.25 > 0 is true between the roots t 1 and t 2 . Thus, the ball's height is greater than 10 meters when t is approximately between 0.47 and 4.02 seconds.
Final Answer The ball's height is greater than 10 meters when t is approximately between 0.47 and 4.02 seconds.
Examples
Understanding projectile motion is crucial in sports like basketball and soccer. When a player kicks a ball, its trajectory can be modeled using quadratic equations, similar to the one in this problem. By analyzing the equation, athletes and coaches can determine the optimal angle and initial velocity required to reach a specific target, such as scoring a goal or making a successful pass. This blend of physics and mathematics enhances performance and strategic decision-making in sports.
